Slow-growth approach: Difference between revisions

Revision as of 06:12, 14 June 2024

The free-energy profile along a geometric parameter $\xi$ can be scanned by an approximate slow-growth approach^[1]. In this method, the value of $\xi$ is linearly changed from the value characteristic for the initial state (1) to that for the final state (2) with a velocity of transformation $\dot{\xi}$ . The resulting work needed to perform a transformation $1 \rightarrow 2$ can be computed as:

{\displaystyle w^{irrev}_{1 \rightarrow 2}=\int_{{\xi(1)}}^{{\xi(2)}} \left ( \frac{\partial {V(q)}} {\partial \xi} \right ) \cdot \dot{\xi}\, dt. }

In the limit of infinitesimally small $\dot{\xi}$ , the work $w^{irrev}_{1 \rightarrow 2}$ corresponds to the free-energy difference between the the final and initial state. In the general case, $w^{irrev}_{1 \rightarrow 2}$ is the irreversible work related to the free energy via Jarzynski's identity^[2]:

{\displaystyle exp^{-\frac{\Delta A_{1 \rightarrow 2}}{k_B\,T}}= \bigg \langle exp^{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T}} \bigg\rangle. }

Note that calculation of the free-energy via this equation requires averaging of the term ${\rm exp} \left \{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T} \right \}$ over many realizations of the $1 \rightarrow 2$ transformation. Detailed description of the simulation protocol that employs Jarzynski's identity can be found in reference ^[3].

Anderson thermostat

For a slow-growth simulation, one has to perform a calcualtion very similar to Constrained molecular dynamics but additionally the transformation velocity-related INCREM-tag for each geometric parameter with STATUS=0 has to be specified. For a slow-growth approach run with Andersen thermostat, one has to:

Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
Set MDALGO=1, and choose an appropriate setting for ANDERSEN_PROB
Define geometric constraints in the ICONST file, and set the STATUS parameter for the constrained coordinates to 0
When the free-energy gradient is to be computed, set LBLUEOUT=.TRUE.

Specify the transformation velocity-related INCREM-tag for each geometric parameter with STATUS=0.

Nose-Hoover thermostat

For a slow-growth approach run with Nose-Hoover thermostat, one has to:

Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
Set MDALGO=2, and choose an appropriate setting for SMASS
Define geometric constraints in the ICONST-file, and set the STATUS parameter for the constrained coordinates to 0
When the free-energy gradient is to be computed, set LBLUEOUT=.TRUE.

Specify the transformation velocity-related INCREM-tag for each geometric parameter with STATUS=0

VASP can handle multiple (even redundant) constraints. Note, however, that a too large number of constraints can cause problems with the stability of the SHAKE algorithm. In problematic cases, it is recommended to use a looser convergence criterion (see SHAKETOL) and to allow a larger number of iterations (see SHAKEMAXITER) in the SHAKE algorithm. Hard constraints may also be used in metadynamics simulations (see MDALGO=11 | 21). Information about the constraints is written onto the REPORT-file: check the lines following the string: Const_coord

References

[woo1997-1] T. K. Woo, P. M. Margl, P. E. Blochl, and T. Ziegler, J. Phys. Chem. B 101, 7877 (1997).

[jarzynski1997-2] C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).

[oberhofer2005-3] . Oberhofer, C. Dellago, and P. L. Geissler, J. Phys. Chem. B 109, 6902 (2005).

[1]

[2]

[3]

@@ Line 8: / Line 8: @@
 can be computed as:
-<math>
+::<math>
 w^{irrev}_{1 \rightarrow 2}=\int_{{\xi(1)}}^{{\xi(2)}}  \left ( \frac{\partial                                      {V(q)}} {\partial \xi} \right ) \cdot \dot{\xi}\, dt.
 </math>
@@ Line 14: / Line 14: @@
 In the limit of infinitesimally small <math>\dot{\xi}</math>, the work <math>w^{irrev}_{1 \rightarrow 2}</math>
 corresponds to the free-energy difference between the the final and initial state.
-In the general case, <math>w^{irrev}_{1 \rightarrow 2}$ </math>is the irreversible work related
+In the general case, <math>w^{irrev}_{1 \rightarrow 2}</math> is the irreversible work related
 to the free energy via Jarzynski's identity<ref name="jarzynski1997"/>:
-<math>
+::<math>
-{\rm exp}\left\{-\frac{\Delta A_{1 \rightarrow 2}}{k_B\,T} \right \}=
+exp^{-\frac{\Delta A_{1 \rightarrow 2}}{k_B\,T}}=
-\bigg \langle {\rm exp} \left \{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T} \right \} \bigg\rangle.
+\bigg \langle  exp^{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T}} \bigg\rangle.
 </math>
@@ Line 31: / Line 31: @@
 == Anderson thermostat ==
-* For a slow-growth simulation, one has to perform a calcualtion very similar to {{TAG|Constrained molecular dynamics}} but additionally the transformation velocity-related {{TAG|INCREM}}-tag for each geometric parameter with <tt>STATUS=0</tt> has to be specified:
+* For a slow-growth simulation, one has to perform a calcualtion very similar to {{TAG|Constrained molecular dynamics}} but additionally the transformation velocity-related {{TAG|INCREM}}-tag for each geometric parameter with <tt>STATUS=0</tt> has to be specified. For a slow-growth approach run with Andersen thermostat, one has to:
 #Set the standard MD-related tags: {{TAG|IBRION}}=0, {{TAG|TEBEG}}, {{TAG|POTIM}}, and {{TAG|NSW}}
 #Set {{TAG|MDALGO}}=1, and choose an appropriate setting for {{TAG|ANDERSEN_PROB}}
-#Define geometric constraints in the {{FILE|ICONST}}-file, and set the {{TAG|STATUS}} parameter for the constrained coordinates to 0
+#Define geometric constraints in the {{FILE|ICONST}} file, and set the '''STATUS''' parameter for the constrained coordinates to 0
 #When the free-energy gradient is to be computed, set {{TAG|LBLUEOUT}}=.TRUE.
@@ Line 61: / Line 61: @@
 </references>
 ----
-[[The_VASP_Manual|Contents]]
-[[Category:Molecular Dynamics]][[Category:Slow-growth approach]][[Category:Theory]][[Category:Howto]]
+[[Category:Molecular dynamics]][[Category:Slow-growth approach]][[Category:Theory]][[Category:Howto]]